\[ \int \frac {\left (1+x^4\right )^{3/4}}{1+2 x^4+2 x^8} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {2-\sqrt {2}}\, \arctan \left (\frac {\sqrt {2-\sqrt {2}}\, x \left (x^{4}+1\right )^{\frac {1}{4}}}{-x^{2}+\sqrt {x^{4}+1}}\right )}{8}+\frac {\sqrt {2+\sqrt {2}}\, \arctan \left (\frac {\sqrt {2+\sqrt {2}}\, x \left (x^{4}+1\right )^{\frac {1}{4}}}{-x^{2}+\sqrt {x^{4}+1}}\right )}{8}+\frac {\sqrt {2-\sqrt {2}}\, \arctanh \left (\frac {\sqrt {2-\sqrt {2}}\, x \left (x^{4}+1\right )^{\frac {1}{4}}}{x^{2}+\sqrt {x^{4}+1}}\right )}{8}+\frac {\sqrt {2+\sqrt {2}}\, \arctanh \left (\frac {\sqrt {2+\sqrt {2}}\, x \left (x^{4}+1\right )^{\frac {1}{4}}}{x^{2}+\sqrt {x^{4}+1}}\right )}{8} \]
command
Integrate[(1 + x^4)^(3/4)/(1 + 2*x^4 + 2*x^8),x]
Mathematica 13.1 output
\[ \frac {1}{8} \left (\sqrt {2-\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )+\sqrt {2+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )+\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )+\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^4\right )^{3/4}}{1+2 x^4+2 x^8} \, dx \]________________________________________________________________________________________